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## Phase detector

The goal of the PLL is to maintain a demodulating sine and cosine that match the incoming carrier. Suppose ${\omega }_{c}$ is the believed digital carrier frequency. We can then represent the actual received carrier frequency as theexpected carrier frequency with some offset, $\stackrel{˜}{{\omega }_{c}}={\omega }_{c}+\stackrel{˜}{\theta }(n)$ . The NCO generates the demodulating sine and cosine with the expected digital frequency ${\omega }_{c}$ and offsets this frequency with the output of the loop filter. The NCO frequency can then be modeled as $\stackrel{^}{{\omega }_{c}}={\omega }_{c}+\stackrel{^}{\theta }(n)$ . Using the appropriate trigonometric identities $\cos A\cos B=1/2(\cos (A-B)+\cos (A+B))$ and $\cos A\sin B=1/2(\sin (B-A)+\sin (A+B))$ . , the in-phase and quadrature signals can be expressed as

${z}_{0}(n)=1/2(\cos (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))+\cos (2{\omega }_{c}+\stackrel{˜}{\theta }(n)+\stackrel{^}{\theta }(n)))$
${z}_{Q}(n)=1/2(\sin (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))+\sin (2{\omega }_{c}+\stackrel{˜}{\theta }(n)+\stackrel{^}{\theta }(n)))$
After applying a low-pass filter to remove the double frequency terms, we have
${y}_{1}(n)=1/2\cos (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))$
${y}_{Q}(n)=1/2\sin (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))$
Note that the quadrature signal, ${z}_{Q}(n)$ , is zero when the received carrier and internallygenerated waves are exactly matched in frequency and phase. When the phases are only slightly mismatched we can use therelation
$\forall \theta , \mathrm{small}\colon \sin \theta \approx \theta$
and let the current value of the quadrature channel approximate the phase difference: ${z}_{Q}(n)\approx \stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n)$ . With the exception of the sign error, this difference is essentially how much we need to offset our NCOfrequency If $\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n)> 0$ then $\stackrel{^}{\theta }(n)$ is too large and we want to decrease our NCO phase. . To make sure that the sign of the phase estimate is right, in this example the phase detector issimply negative one times the value of the quadrature signal. In a more advanced receiver, information from boththe in-phase and quadrature branches is used to generate an estimate of the phase error. What should the relationship between the I and Q branches be fora digital QPSK signal?

## Loop filter

The estimated phase mismatch estimate is fed to the NCO via a loop filter, often a simple low-pass filter. For thisexercise you can use a one-tap IIR filter,

$y(n)=\beta x(n)+\alpha y(n-1)$
To ensure unity gain at DC, we select $\beta =1-\alpha$

It is suggested that you start by choosing $\alpha =0.6$ and $K=0.15$ for the loop gain. Once you have a working system, investigate the effects of modifying these values.

## Matlab simulation

Simulate the PLL system shown in [link] using MATLAB. As with the DLL simulation, you will have to simulate the PLL on a sample-by-sample basis.

Use [link] to implement your NCO in MATLAB. However, to ensure that the phase term does not grow toinfinity, you should use addition modulo $2\pi$ in the phase update relation. This can be done by setting $\theta (n)=\theta (n)-2\pi$ whenever $\theta (n)> 2\pi$ .

[link] illustrates how the proposed PLL will behave when given a modulated BPSK waveform. In this case thetransmitted carrier frequency was set to $\stackrel{˜}{{\omega }_{c}}=\frac{\pi }{2}+\frac{\pi }{1024}$ to simulate a frequency offset.

Note that an amplitude transition in the BPSK waveform is equivalent to a phase shift of the carrier by $\frac{\pi }{2}$ . Immediately after this phase change occurs, the PLL begins to adjust the phase to force the quadraturecomponent to zero (and the in-phase component to $1/2$ ). Why would this phase detector not work in a real BPSK environment? How could it be changed to work?

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